sign in sign up
<<
Home , Particle physics

QCD, jets and VII. QCD, jets and gluons (QCD)

Quantum Chromodynamics (QCD) is the theory of strong interactions.

Interactions are carried out by a massless spin-1 particle – gauge

In (QED) gauge are , in QCD – gluons

Gauge bosons couple to conserved charges: photons in QED – to electric charges, and gluons in QCD – to colour charges

The acts the same on u,d,s,c,b and t because the strong interaction is flavour-independent.

Oxana Smirnova & Vincent Hedberg Lund University 143 QCD, jets and gluons Particle Physics Gluons carry colour charges themselves !

C C u u C I = 1/2 Y = 1/3 C _ 3 I3 = 0 Y = 2/3

C C I3 = 1/2 Y = 1 C C _ C I = 0 Y = 2/3 C 3 s s I3 = 1/2 Y = 1/3

Figure 58: exchange between quarks.

The colour quantum numbers of the gluon in the figure above are:

1 IC ==IC() – IC()b --- (86) 3 3 3 2 YC ==YC()r – YC()b 1

C C Gluon colour χC = r g I = 1 Y = 0 wavefunctions: g1 3 C χC = r g I = 1 Y C = 0 g2 3 C C χC = r b I = 1/2 Y = 1 g3 3 C χC = r b I = 1/2 Y C = 1 g4 3 C C χC = g b I = 1/2 Y = 1 g5 3 C χC = g b I =1/2 Y C = 1 g6 3 C C χC 1/ 2 ( g g - r r ) I = 0 Y = 0 g7 = 3 C C χC 1/ 6 ( g g - r r - 2 b b ) I = 0 Y = 0 g8 = 3

Oxana Smirnova & Vincent Hedberg Lund University 144 QCD, jets and gluons Particle Physics

Gluons can couple to other gluons since gluons carry colour !

Figure 59: Lowest-order contributions to gluon-gluon .

All observed states have zero colour charge - colour confinement.

Gluons does not exist as free since they have colour charge.

Bound colourless states of gluons are called (not detected experimentally yet).

Oxana Smirnova & Vincent Hedberg Lund University 145 QCD, jets and gluons Particle Physics The principle of : At short distances the strong interactions are weaker ⇒ quarks and gluons are essentially free particles ⇒ the interaction can be described by the lowest order diagrams.

At large distances the strong interaction gets stronger ⇒ the interaction can be described by high-order diagrams.

The -antiquark potential is: α 4 s Vr()= –------(r< 0, 1 fm ) 3 r Vr()= λr (r≥ 1fm ) α where s is the strong constant and r the distance between the quark and the antiquark.

Due to the complexity of high-order diagrams, the very process of confinement can not be calculated analytically ⇒ only numerical models are available.

Oxana Smirnova & Vincent Hedberg Lund University 146 QCD, jets and gluons Particle Physics

P1 P2

q = P1 P2

P4 = P3 + q P3 where P and q are -momentum four vectors: P = (E,p) = (E,px,py,pz) , q = (Eq,q) = P 1 P2= (E1 - E2 P1 P2) the momentum and energy transfer is: | q = | q |= P1 P2| ν = Eq = E1 - E2 (where E and P are in the restframe of particle 3). The energy-momentum transfer is given by: 2 2 Q = q .q = ()P1 P2 Q2 2 2 2 = q . q Eq = (P1 P2 ) - (E1 - E2) This can be regarded as the invariant of the exchanged since the squared mass of a particle is given by M2 = P . P The of the is given by: 2 . 2 W = P4 P4 = (P3 + q )

Oxana Smirnova & Vincent Hedberg Lund University 147 QCD, jets and gluons Particle Physics The strong coupling constant

α The strong coupling constant s is the α analogue in QCD of em and it is a measure of the strength of the interaction. α s is not a true constant but a “running constant” since it decreases with increasing Q2. α In leading order of QCD, s is given by

12π α = ------(87) s ()()2 ⁄ Λ2 33– 2Nf ln Q

Here Nf is the number of allowed quark flavours, and Λ≈0.2 GeV is the QCD scale parameter which has to be defined experimentally.

Oxana Smirnova & Vincent Hedberg Lund University 148 QCD, jets and gluons Particle Physics

0.5 Theory Data

α NLO NNLO s(Q) Lattice Deep Inelastic Scattering e+e- 0.4 Collisions Heavy Quarkonia

Λ(5) α (Μ ) MS s Z 275 MeV 0.123 QCD 0.3 0.119 α 4 ) { 220 MeV O( s 175 MeV 0.115

0.2

0.1

1 10 100 Q [GeV]

Figure 60: The running of the strong coupling constant.

Oxana Smirnova & Vincent Hedberg Lund University 149 QCD, jets and gluons Particle Physics - annihilation A clean process with which to study QCD is: ∗ e+ + e- → γ → hadrons (88)

+ e q

jets of hadrons γ∗ e- q

Figure 61: e+e- annihilation into hadrons. −

Figure 62: An e+e- annihilation in which two quark jets were created (the event was recorded by the JADE experiment at DESY).

Oxana Smirnova & Vincent Hedberg Lund University 150 QCD, jets and gluons Particle Physics

In the lowest order e+e- annihilation process, a or a Z0 is produced which converts into a quark-antiquark pair.

The quark and the antiquark fragment into observable hadrons.

Since the quark and antiquark momenta are equal and counterparallel, hadrons are produced in two jets of equal going in opposite direction.

The direction of a reflects the direction of the corresponding quark.

Oxana Smirnova & Vincent Hedberg Lund University 151 QCD, jets and gluons Particle Physics The total cross-section of e+e- → hadrons is often expressed as: σ()e+e- → hadrons R ≡ ------σ()e+e- → µ+µ- The for production and hadron production is the same if the number of quark flavours and colours are taken into account:

2 R = NcΣ eq

Here Nc = 3 is the number of colours and eq is the charge of the quarks.

If sm< ψ then

2 22 2 2 2 R =Nc(eu+ed+es ) = 3 ((-1/3)+(-1/3)+(2/3)) = 2

< 22 2 2 If smϒ then R =Nc(eu+ed+es +ec)=10/3 and

> 22 2 2 2 If smϒ then R =Nc(eu+ed+es +ec+eb)=11/3

Oxana Smirnova & Vincent Hedberg Lund University 152 QCD, jets and gluons Particle Physics If the of hard gluons is taken into α account, an extra factor proportional to s has to be added: α ()Q2 R3e= ∑ 21 + ------s - qπ q

Nc Nc

Nc No colour 2 2 2 2 e2+e2+e2 2 2 eu+ed+es+ec u d s+ec+eb 2 2 2 eu+ed+es

Figure 63: The measured R-value and the predicted R-value for different theoretical assumptions.

Oxana Smirnova & Vincent Hedberg Lund University 153 QCD, jets and gluons Particle Physics A study of the angular distribution of the jets give about the spin of the quarks.

The angular distribution of the process ∗ e+ + e- → γ → µ+ + µ- is given by: dσ πα2 2 ------()e+e- → µ+µ- = ------()1 + cos θ dcosθ 2Q2 where θ is the production angle with respect to the direction of the colliding .

If quarks have spin 1/2 they should have the following angular distribution: dσ 2πα2 2 ------()e+e- → qq = N e ------()1 + cos θ dcosθ c q2Q2 where eq is the fractional charge of a quark and Nc = 3 is the number of colours.

If quarks have spin 0 the angular distribution should be: dσ 2πα2 2 ------()e+e- → qq = N e ------()1 – cos θ dcosθ c q2Q2

Oxana Smirnova & Vincent Hedberg Lund University 154 QCD, jets and gluons Particle Physics

Figure 64: The angular distribution of the quark jet in e+e- , compared with models.

The experimentally measured angular dependence of jets is clearly following(1+cos2θ) ⇒ jets are associated with spin-1/2 quarks.

Oxana Smirnova & Vincent Hedberg Lund University 155 QCD, jets and gluons Particle Physics If a high-momentum (hard) gluon is emitted by a quark or antiquark, it fragments to a jet of its own, which leads to a three-jet event:

+ e q

g jets of hadrons γ∗ e- q

Figure 65: A three-jet event in an e+e-annihilation as seen by the JADE experiment.

Oxana Smirnova & Vincent Hedberg Lund University 156 QCD, jets and gluons Particle Physics

The probability for a quark to emit a gluon is α proportional to s and by comparing the rate of α two-jet and three-jet events one can determine s. α ± s=0.15 0.03 for ECM=30 to 40 GeV

Figure 66: The principal scheme of hadron production in e+e- annihilations. Hadronization (= fragmentation) begins at distances of order 1 fm between the partons.

Oxana Smirnova & Vincent Hedberg Lund University 157 QCD, jets and gluons Particle Physics By measuring angular distributions of jets one can confirm models where gluons are spin-1 bosons. This is done by measuring: θ sinθ φ sin 2- 3 cos = θ sin 1 where the angles are described below:

θ Jet 2 Jet 1 3 Emax θ θ 1 2 Jet 3 Emin

gluon spin = 0

gluon spin = 1

Figure 67: An angular distribution of jets compared to QCD calculations with a spin 0 and a spin 1 gluon.

Oxana Smirnova & Vincent Hedberg Lund University 158